lncs 7370 - graph minors and parameterized algorithm...

Graph Minors and Parameterized

Algorithm Design�

Dimitrios M. Thilikos

Department of Mathematics, National and Kapodistrian University of Athens,Panepistimioupolis, GR-15784, Athens, Greece

[email protected]

Abstract. The Graph Minors Theory, developed by Robertson and Sey-mour, has been one of the most influential mathematical theories in pa-rameterized algorithm design. We present some of the basic algorithmictechniques and methods that emerged from this theory. We discuss itsdirect meta-algorithmic consequences, we present the algorithmic appli-cations of core theorems such as the grid-exclusion theorem, and we givea brief description of the irrelevant vertex technique.

Keywords: graph minors, parameterized algorithms, treewidth, bidi-mensionality, irrelevant vertex technique, linkages.

1 Introduction

Graph Minors Theory (GMT) was developed by Robertson and Seymour in aseries of 23 papers, between 1984 and 2009. Among them, the second paper ofthe series was published in the Journal of Algorithms while all the rest were pub-lished in the Journal of Combinatorial Theory Series B. The main theoreticalachievement of this project was the proof of Wagner’s conjecture, now known asthe Robertson & Seymour Theorem, stating that graphs are well-quasi-orderedunder the minor containment relation. Besides its purely mathematical impor-tance, GMT induced a series of powerful algorithmic results and techniques thathad a deep influence on theoretical computer science. More particularly, GMThas been one of the most powerful “mathematical engines” in the theory anddesign of parameterized algorithms. In particular, a considerable part of the ba-sic techniques in parameterized algorithm design is directly or indirectly linkedto results from GMT. Moreover, GMT offered the theoretical base for the un-derstanding and resolution of some of the most prominent graph-algorithmicproblems in parameterized complexity. In what follows, we give a brief presen-tation of the main results and techniques in this area.

� This research has been co-financed by the European Union (European Social Fund– ESF) and Greek national funds through the Operational Program “Educationand Lifelong Learning” of the National Strategic Reference Framework (NSRF) -Research Funding Program: “Thalis. Investing in knowledge society through theEuropean Social Fund”.

H.L. Bodlaender et al. (Eds.): Fellows Festschrift, LNCS 7370, pp. 228–256, 2012.c© Springer-Verlag Berlin Heidelberg 2012

Graph Minors and Parameterized Algorithm Design 229

Our presentation is organized as follows. In Section 2 we give the definitionsof some basic combinatorial and algorithmic concepts. In Section 3.1 we presentthe main algorithmic consequences of the GMT, mainly from the parameterizedcomplexity viewpoint. Section 4 is devoted to the celebrated grid-exclusion theo-rem and its applications to bidimensionality theory. Finally, Section 5, attemptsa short presentation of the irrelevant vertex technique and its applications.

2 Basic Definitions

All graphs we consider are finite, undirected and simple, i.e., they do not havemultiple edges or loops. Given a graph G we denote by V (G) and E(G) itsvertex and edge set respectively. The size (reps. magnetite) of a graph G is thenumber of its vertices (reps. edges) and is denoted by n(G) (reps. m(G)), i.e.,n(G) = |V (G)| (m(G) = |E(G)|). We denote by G \ v the graph obtained byremoving v (along with its incident edges) from G. The neighborhood of a vertexv ∈ V (G), denoted by NG(v), is the set of edges in G that are adjacent to v.The degree of a vertex v ∈ V (G) is the cardinality of its neighborhood in G.We denote by Kr the complete graph on r vertices and by Kr,q the completebipartite graph with r vertices in its one part and q in the other. Finally, wedenote by Gk the (k× k)-grid, i.e., the Cartesian product of two paths of lengthk − 1 (see Figure 1).

Fig. 1. The (11, 11)-grid G11

2.1 Relations on Graphs and Obstructions

We say that a graph H is a subgraph of a graph G if H can be obtained by G byremoving edges or vertices. The contraction of an edge e = {x, y} from G is theremoval from G of all edges incident to x or y and the insertion of a new vertexve that is made adjacent to all the vertices of (NG(x) \ {y}) ∪ (NG(y) \ {x}).Given two graphs H and G, we say that H is a contraction of G, denoted byH ≤c G, if H can be obtained from G by a (possibly empty) series of edgecontractions.

230 D.M. Thilikos

G H

v

Fig. 2. The graph H is the result of the contractions of the bold edges in G to thevertex v

H is a minor of G if H is a contraction of some subgraph of G. A graph H isa topological minor of G (denoted by H ≤t G) if G contains as a subgraph somesubdivision of H (a subdivision of a graph H is any graph obtained by replacingsome of its edges by paths between the same endpoints). Given a partial orderingrelation ≤ on graphs, we say that a graph class G is closed under ≤ if for everyG ∈ G, H ≤ G implies that H ∈ G. Let G be a graph class that is closed underthe minor relation. An ≤-anti-chain is a set of graphs that are pairwise non-comparable with respect to ≤. For example the set of graphs A = {K2,r | r ≥ 2}is a ≤c-antichain but not an ≤m-antichain or a ≤t-antichain.

We define the ≤-obstruction set of a graph class G, denoted by obs≤(G), asthe set of all ≤-minimal graphs that do not belong to G. Clearly, by definition,the ≤-obstruction set of a graph class is an ≤-anti-chain. Obstruction sets can beseen as alternative characterizations of graph classes and, in many cases, reveal agood deal from their structural characteristics. For example, it easy to verify thatobs≤m(T ) = obs≤t(T ) = {K3}, obs≤m(O) = obs≤t(O) = {K4, K2,3}, whereT and O are the classes of all acyclic and all outerplanar graphs respectively,and obs≤c(O∗) = {K4, K2,3, K

+2,3} where O∗ are the connected outerplanar

graphs and K+2,3 is the graph obtained from K5 by removing a triangle. The

most classic theorems on obstruction characterization of graph classes are theKuratowski-Pontryagin’s theorem [87] and Wagner’s theorem [121], stating thatobs≤m(P) = {K5, K3,3} and obs≤t(P) = {K5, K3,3} respectively, where P isthe class of all planar graphs.

2.2 Parameterized Problems and Algorithms

The idea of problem parameterization is to treat algorithmic problems as param-eterized entities and to evaluate the complexity of the corresponding algorithmsby considering the way parameters appear in their running times. As here wedeal with problems on graphs, we adapt the classic definitions of parameterizedcomplexity to the case where problem inputs represent graphs.

Formally, a parameterized problem on graphs is a subset Π of Σ∗ × N, whereΣ is some alphabet and, in each (I, k) ∈ Σ∗ × N, I encodes a combinatorialstructure related to one, or more, graphs. For this, we agree that n (resp. m) isthe maximum size (resp. magnitude) of the graphs encoded in I and we insistthat |(I, k)| = O(m). We call I the main part of the input and we say thatk is the parameter of the problem. Two instances (I, k), (I ′, k′) ∈ Σ∗ × N areequivalent with respect to Π if (I, k) ∈ Π ⇐⇒ (I ′, k′) ∈ Π .


We say that Π is fixed parameter tractable if there exists a function1 f : N→ N

and an algorithm deciding whether (I, k) ∈ Π in O(f(k) · nc) steps, where c isa constant not depending on the parameter k of the problem. We call such analgorithm FPT-algorithm or, to express concretely the choice of f and c, wesay that Π ∈ O(f(k) · nc)-FPT. A parameterized problem on graphs belongs tothe parameterized class FPT if it can be solved by an FPT-algorithm. In fact,not all parameterized problems belong to the class FPT. There is a hierarchy ofparameterized complexity classes, namely

FPT ⊆W[1] ⊆W[2] ⊆W[3] ⊆ . . . ⊆W[SAT ] ⊆W[P] ⊆ XP,

and appropriate parameter-preserving reductions such that, when a problem ishard for some of them (other than FPT), it is not expected to have an FPT-algorithm (all inclusions in this hierarchy are believed to be strict). See themonographs [39, 44, 97] for more details on parameterized complexity.

Time bounds for parameterized algorithms have two parts. The term f(k) iscalled parameter dependence and, is typically a super-polynomial function. Onthe other hand, the term nc is a polynomial function and we call it polynomialpart. In most of the problems that we examine here, I will encode a simple graph.To simplify notation, we frequently write “Ok(nc)” instead of “f(k) · (nc)” forsome recursive function f : N→ N” and, in this case, we refer to the function fhidden in the Ok notation as the parameter dependence.

3 Algorithmic Consequences of the GMT

3.1 Well-Quasi-Ordering

The main combinatorial result of the GMT fits in the more general framework ofthe theory of Well-Quasi-Orderings, first developed by Graham Higman2 underthe name “finite basis property” [65]. Given a set X and a partial ordering ≤on X we say that X is well-quasi-ordered under ≤ if none of its subsets is aninfinite ≤-antichain.

Theorem 1 (Robertson & Seymour Theorem [110]). The set of all graphsis well-quasi-ordered under minors.

In other words, Theorem 1 says that If G is an infinite set of graphs then there ex-ist two graphs H, G ∈ G such that H is a minor of G. The proof of theorem 1 wasconcluded in paper XX of the Graph Minors Series. Before its proof, the state-ment of Theorem 1 was known as Wagner’s conjecture. However, as mentionedby Diestel in [32], Wagner said that he had never made such a conjecture. A sim-ilar conjecture, on the well-quasi-ordering of trees under the topological minor1 Notice that in the definition of FPT f is not necessarily a recursive function.2 As mentioned in [86], the same theory was also developed in some unpublished

manuscript of Erdos and Rado, while its first hints can be traced back to B. H. Neu-mann [96]

232 D.M. Thilikos

relation was made by Vazsonyi and was proved in 1960 independently3 by JosephKruskal and S. Tarkowski [91]. Interesting results on the meta-mathematics ofKruskal’s tree theorem as well as Roberson & Seymour’s theorem can be foundin [55] and [54] respectively.

Consider the following parameterized problem:

H-Minor Checking

Instance: Two graphs G and H .Parameter: k = |V (H)|.Question: Is H a minor of G?

The main algorithmic contribution of the GMT is the following result.

Theorem 2 (Robertson and Seymour[108]). One can construct an algo-rithm that, given a n-vertex graph G and a k-vertex graph H, checks whether His a minor of a graph G in Ok(n3) steps. In other words, H-Minor Checking ∈Ok(n3)-FPT.

Actually, Robertson and Seymour in [108] describe an Ok(n3)-step algorithmthat solves a generalization of the H-Minor Checking and another celebratedproblem, namely the k-Disjoint Paths problem. In Section 5, we give a roughdescription of the main ingredients of the algorithm in Theorem 2 especially forthe k-Disjoint Paths problem. Recently, this running time was improved to aquadratic one for the k-Disjoint Paths problem in [74].

The good news about Theorem 2 is that it is constructive (contrary to The-orem 1) and there is a recursive function hidden in the Ok notation. The badnews is that, according to the algorithm in [108] and the proof of its correctnessin [111] and [107], the values of this function are immense4, even for small valuesof k. David Johnson mentioned in [67]:

“for any instance G = (V, E) that one could fit into the known universe,one would easily prefer |V |70 to even constant time, if that constant hadto be one of Robertson and Seymour’s”.

Moreover, in [67], David Johnson estimates that just one constant in the param-eter dependence of Theorem 2 is roughly

2↑222

2↑2↑Θ(r)

where 2↑r denotes a tower 222. . .

involving r 2’s. Clearly, such type of constantsmay create reasonable doubts to computer scientists on whether such an algo-rithm may be considered to be an “algorithm” of some practical meaning. In fact,to investigate until which point these constants can be improved is an open andchallenging problem in parameterized complexity and algorithms (see e.g. [3]).

3 A shorter and quite elegant proof of Vazsonyi’s conjecture was given by Nash-Williams in 1963 [95].

4 Perhaps the word “immense” is somehow moderate here. Instead, Fellows andLangston used the expression “mind-boggling” in [43].


3.2 Minor-Closed Graph Parameters

A parameter on graphs (or a graph parameter) is any function that maps graphsto integers and with the property that it is invariant under graph isomorphism.Let ≤ be a relation on graphs. We say that a graph parameter p is closed under≤ (or, simply, ≤-closed ) if for every two graph H and G, H ≤ G implies thatp(H) ≤ p(G). We define the ≤-obstruction family of p as the parameterizedgraph class

O≤p,k = obs≤({G | p(G) ≤ k}).

Consider the following parameterized meta-problem.

k-Parameter Checking for pInstance: a graph G and an integer k ≥ 0.Parameter: kQuestion: p(G)≤ k?

Theorems 1 and 2 together have the following dramatic consequence.

Theorem 3. For every parameter p that is closed under minors there exists analgorithm that solves the problem k-Parameter Checking for p in f(k) · n3

steps for some function f .

Proof. Recall that, by definition, no two graphs in O≤m

p,k can be comparablegraphs under the minor relation. It follows, from Theorem 1, that O≤m

p,k is afinite set. Let g(k) = |O≤m

p,k |. As p is closed under minors, it holds that

p(G) ≤ k ⇐⇒ ∀H ∈ O≤m

p,k H �≤m G.

Therefore, to check whether p(G) ≤ k it is enough to apply g(k) times theOk(n3) step algorithm of Theorem 2 and check whether some member of O≤m

p,k

is contained as a minor in G.

Theorem 3 had a great impact in parameterized complexity as it implied amassive classification of problems in the class FPT. In that sense, Theorem 3is an algorithmic meta-theorem because it provides a generic condition (minor-closedness) for a parameterized problem that automatically implies the existenceof an FPT-algorithm for it. Unfortunately, the proof of Theorem 1 does notprovide any general “meta-algorithm” to compute the set O≤m

p,k and, that way,construct the claimed algorithm for each p. In fact, due to the meta-mathematicsof Theorem 1 [54], such an meta-algorithm does not exist. As observed in [43],there is no algorithm that, given a Turing machine accepting precisely the graphsof a minor-closed graph class F , outputs obs≤m(F) (see also [119]). However,Theorem 3 gave important (mathematical) energy to Parameterized Algorithmsas it acted as an “encouraging factor”. The knowledge that an algorithm existsfor a specific problem, induces the challenge to construct one and, in a sense,provides the courage to try to accomplish such a task.

234 D.M. Thilikos

In order to cope with the inherent non-constructivity of Theorem 3 one maystudy specific parameters where the computation of the set O≤m

p,k (or, at least, ofsome upper bound to the function g(k)) in the proof of Theorem 3 is possible.However, this is not an easy task, even for simple parameters. According to [34],if the problem of checking whether p(G) ≤ k is NP-complete, then |O≤m

p,k | is asuper-polynomial function of k, unless the polynomial hierarchy collapses to ΣP

3 .Characterizations of p(G) ≤ k (yielding better lower bounds for |O≤m

p,k |) havebeen provided for several parameters [10, 20, 40, 58, 84, 98, 99, 115, 117, 118].However, to our knowledge, there is not yet a natural parameter p for which acomplete characterization of O≤m

p,k is known. A more promising strategy towardsdetecting constructive fragments of Theorem 3, is to detect parameters – orfamilies of parameters – where O≤m

p,k is recursive. For this one may either proveupper bounds for |O≤m

p,k |, as done in [57] for the case of branchwidth5, or providepartial characterizations of O≤m

p,k , as done in [2, 19, 21, 89, 94], that permit itsrecursive computation.

At this point, we should mention that all theorems of this section have theircounterparts in another partial relation on graphs, the one of immersion. Thelift of two incident edges is the operation of removing two edges e1 = {x, y}and e2 = {x, z} (incident to a common vertex x) and adding the edge {y, z}.We say that a graph H can be immersed in a graph G, denoted by H ≤im G,if H can be obtained from a subgraph of G by a (possibly empty) sequence ofedge lifts. According to the last paper of the Graph Minor series [112], graphsare well-quasi-ordered under immersions, i.e., Theorem 1 holds also if we replaceminors by immersions. Therefore, in order to prove a counterpart of Theorem 3for the case of immersion-closed parameters, we need an algorithm that given ann-vertex graph G and a k-vertex graph H , checks where H ≤im G in Ok ·(n(G))3

steps. Recently, a construction of such an algorithm was given in [61]. This makesit possible to derive the following meta-algorthmic result.

Theorem 4. If p is a parameter that is closed under immersions, then thereexists an algorithm that solves the problem k-Parameter Checking for p inf ′(k) · n3 steps for some function f ′.

In fact, the main result of [61] proves the FPT membership of topological minortesting, i.e., given two graphs H and G, check whether H ≤t G (the parameteris the size of H). This means that there is a counterpart of Theorem 2 for thetopological minor relation as well. This might create some hope that Theorem 3holds for topological minors as well. Unfortunately, this requires an analogue ofthe combinatorial Theorem 1 which does not exist as it is possible to constructan infinite class of graphs that are pairwise non-comparable with respect tothe topological minor relation: just take all cycles with their edges duplicated.

5 Branchwidth was introduced in the paper X of the Graph Minor Series [106] and,from that point and then, was used as an alternative for treewidth (defined formallyin Section 4). Treewidth and branchwidth can be seen as twin parameters, as theone is a constant factor approximation of the other.


An other argument for the non-existence of analogues of Theorems 1 and 3 fortopological minors is given by the Topological Bandwidth problem askingwhether the topological bandwidth of a graph is at most k. The topologicalbandwidth of a graph G is denoted by tbw(G) and is defined as

tbw(G) = min{k | ∃q ≥ 1: G ≤t P kq }

(P kq is obtained by a path Pq of length q if we make adjacent any two vertices of

distance ≤ k in Pq). It is easy to observe that tbw is closed under topologicalminors. In [42] it is mentioned that Topological Bandwidth is W[t]-hard forall t ≥ 1 – the proof is a modification of the proof for the case of Bandwidth

in [15]. This implies that, under reasonable assumptions in parameterized com-plexity theory, the anti-chain corresponding to the ≤t-obstruction family O≤t

tbw,k

is infinite for an infinite set of values of k.

4 Grid-Exclusion and Bidimensionality

4.1 Treewidth

Treewidth has been one of the main contributions of GMT to algorithmic graphtheory. While, as a concept, its indices can be traced back to the work of Gavrilin [56], its formal birth as a graph parameter occurred in the second paper of theGraph Minors series [104]. Currently, there are at least six equivalent definitionsof tree-width. We present the original one from [104].

A tree decomposition of a graph G is a pair (X , T ) where T is a tree andX = {Xi | i ∈ V (T )} is a collection of subsets of V (G) such that:

1.⋃

i∈V (T ) Xi = V (G);2. for each edge {x, y} ∈ E(G), {x, y} ⊆ Xi for some i ∈ V (T ), and3. for each x ∈ V (G) the set {i | x ∈ Xi} induces a connected subtree of T .

The width of a tree decomposition ({Xi | i ∈ V (T )}, T ) is maxi∈V (T ) {|Xi| − 1}.The treewidth of a graph G, denoted by tw(G), is the minimum width over alltree decompositions of G.

If, in the above definitions, we restrict the tree T to be a path then we definethe notions of path decomposition and pathwidth. We write pw(G) to denote thepathwidth of a graph G. Pathwidth was defined earlier than treewidth in thefirst paper is the Graph Minors Series [102].

Treewidth can intuitively be seen as a measure of the topological resemblanceof a graph to a tree or, alternatively, as a measure of the “global connectivity”of a graph. Similarly, pathwidth can be seen as a measure of the topologicalresemblance of a graph to a path.

Counting Monadic Second Order Logic (CMSOL) is a logic on graphs6 wherethe domain is the set of vertices and edges, there are predicates for vertex-vertex6 We should stress that CMSOL is not only a logic on graphs but also on more general

combinatorial objects called strucures.

236 D.M. Thilikos

adjacency and edge-vertex incidence, there is quantification over edges, vertices,edge sets and vertex sets, and there is a predicate Cardr,p(S) which expresseswhether the size of a set S is r modulo p.

The importance of treewidth for algorithmic graph theory is illustrated by thecelebrated Courcelle’s theorem stating that if Πk is a parameterized propertyof graphs expressible by a CMSOL formula φk, then there is an algorithm that,given as input a graph G, can check whether G satisfies property Πk (i.e., whetherG ∈ Πk) in O|φk|+tw(G)(n) steps. Moreover, there exists a meta-algorithm that,given φk, outputs such an algorithm. A proof of Courcelle’s theorem can befound in [39, Chapter 6.5] and [44, Chapter 10] and similar results appeared byArnborg, Lagergren, and Seese in [8] and Borie, Parker, and Tovey in [17]. Analternative game-theoretic proof has appeared recently in [81, 82].

Courcelle’s theorem had a deep influence in parameterized algorithms as itautomatically yields FPT-algoriths for a wide family of problems, provided thatthe treewidth of their instances is bounded by a function of the parameter k. Thenatural challenge is whether and when a parameterized problem can be reducedto its bounded treewidth variant. For this, an important step is to detect whatkind of combinatorial structures are contained in a graph with big treewidth.The most prominent structure of this type is the grid Gk. Let gm(G) be themaximum k for which G contains Gk as a minor. A valuable theoretical tool inthis direction was given by the following result of the GMT.

Theorem 5 ([105]). There exists a recursive function f : N → N such thattw(G) ≤ f(gm(G)).

While the above result appeared in the fifth paper of the series, a preliminaryvariant of it, where G is planar, appeared earlier in [103]. As every graph con-taining Gk as a minor has treewidth at least k, Theorem 5 implies that tw andgm are parametrically equivalent: a bound to the one of them implies a boundto the other. The initial estimation of the parameter dependence in Theorem 5was huge. However, a better one appeared in [113] where it was proven thattw(G) = 202·(gm(G))5 . An alternative, and relatively simpler, proof of Theo-rem 5 was given in [33]. To see the use of Theorem 5 in parameterized algorithmdesign, consider a parameter p that satisfies the following properties:

i. p is closed under taking of minors.ii. there exists a recursive function t : N→ N such that p(Gt(k)) > k for every

non-negative integer k.iii. One can construct an algorithm that, given a tree-decomposition of G of

width at most q and an integer k, checks whether p(G) ≤ k in l(k, q) · nO(1)

steps for some recursive function l : N× N→ N.

Clearly, the first two conditions are easy to check for most instantiations of p.Moreover, the third one follows directly from Courcelle’s theorem if for eachk, Πk = {G | p(G) ≤ k} is expressible by a CMSOL formula φk. There aremany examples of such parameters. Typical examples are the vertex cover of agraph, i.e., the minimum number of vertices that meets all vertices of G and the


feedback vertex set of a graph, i.e., the minimum number of vertices meeting allcycles of G. A direct consequence of Theorem 5 is the following (constructive)special case of Theorem 3:

Lemma 1. Let p be a parameter satisfying conditions i–iii above for some t andl. Then it is possible to construct an algorithm that, given as input a graph Gand an integer k, checks whether p(G) ≤ k in (2O(f(t(k)) + l(k, 4 ·f(t(k)))) ·nO(1)

steps where f is the function in Theorem 5.

Proof. The algorithm in Lemma 1 works as follows: First of all, it uses an FPT-approximation algorithm for treewidth, i.e., an algorithm that given a graph Gand an integer q, either outputs a tree decomposition of G of width at most α · qor reports that tw(G) > q in z(q) · nβ steps. Various algorithms of this typehave been proposed in [7, 14, 88, 100, 108] for different trade-offs between z,α, and β. Among them, we pick the one form [7] where z(q) = 24.38·q, α = 4,and β = 2. We run this algorithm for G and q = f(t(k)). If it outputs a treedecomposition of width ≤ 4 · f(t(k)) then we use the algorithm of Property iiiand solve the problem in l(k, 4·f(t(k)))·nO(1) steps. If the algorithm reports thatthe treewidth of G is more than f(t(k)), then from Theorem 5, G contains Gt(k)

as a minor. In such a case, the algorithm directly outputs a negative answer as,from Properties i and ii, p(G) ≥ p(Gt(k)) > k.

The idea of the above proof is also known as the Win/win approach: we eitherhave an answer to the problems directly because the treewidth is big enoughor we solve the problems use dynamic programming on a tree-decomposition ofbounded width.

Clearly, the running time of the algorithm in Lemma 1 depends on the func-tions f, t, and l. In what follows, we comment on the current bounds on eachone of them.

l: As we have already mentioned, the (constructive) existence of l, follows fromCourcelle’s theorem for the wide family of problems that are expressible inCMSOL. However, the bounds on l, derived from the proof, are huge and thismay dismiss any hope for a good parameter dependence (see [53]). However,for many problems it is possible to directly apply dynamic programing onthe tree decomposition and derive moderate bounds on l such as l(k, q) =2O(q2)·kO(1), or l(k, q) = 2O(q log q)·kO(1) or, even better, l(k, q) = 2O(q)·kO(1).Clearly, time bounds of the third type are more attractive. For this reason,we say that a parameter p is single exponentially solvable with respect totreewidth if there exists an algorithm that, given G and k, checks whetherp(G) ≤ k in 2O(tw(G)) · nO(1) steps. There is a quite extended bibliographyon how to do fast dynamic programming on graphs of bounded treewidth;as a sample of this, we just mention [5, 6, 9, 11, 13, 16, 22, 35, 35, 36, 37,37, 38, 114, 120, 120].

t: Bounds are much better for the function t. For most natural graph param-eters, it holds that t(k) = O(k) while for some of them, including tw andpw, it holds that t(k) = Θ(k). However, there is a wide family of parameters

238 D.M. Thilikos

where t(k) = O(√

k). This intuitively says that a certificate for the valueof such a parameter spreads “bidimensionally” inside a (k × k)-grid. For in-stance any vertex cover of Gk should have size at least k ·�k

2 � = Ω(k2) as thevertices of such a set should cover edges all over the “area” of the grid. Sim-ilarly, a feedback vertex set of Gk should have size at least (�k

2 �)2 = Ω(k2)as the vertices of such a set should cover all (�k

2 �)2 members of a packing of“squares” in Gk.If such a parameter is also closed under taking of minors then we call itminor bidimensional.

f : To improve the function f , i.e., the parameter dependence in Theorem 5,is an important challenge as, even for the parameters with most moderateinstantiations of l and t, k-Parameter Checking for p could be only clas-

sified in 22kO(1)

· nO(1)-FPT. Robertson, Seymour, and Thomas conjecturedin [113] that f can be a polynomial function. This would directly imply thatk-Parameter Checking for p belongs to 2kO(1) · nO(1)-FPT for a widefamily of parameters (see [30] for more discussions and conjectures on thisissue).Another interesting problem is to lower bound the contribution of f in The-orem 5. As mentioned by Robertson, Seymour, and Thomas in [113] thereare graphs excluding Gk as a minor that have treewidth Ω(k2 · log k). Tosee this, one may use the result in [23] (see also [41, 122]) to construct anO(1)-regular Ramanujan graph G on n vertices that has girth Ω(log n). Onecan easily verify that gm(G) = O(

√n

log n ). The claimed bound follows becauseRamanujan graphs are expanders and thus tw(G) = Ω(n). It is a challengingquestion whether any bound better than this one can be proven.Towards achieving a polynomial dependance between treewidth and the sizeof an excluded grid, Reed and Wood defined in [101] the notion of a grid-like-minor. A grid-like-minor of order k in a graph G is a set of paths in Gwhose intersection graph is bipartite and contains a Kk as a minor. Clearly,the rows and columns of the (k×k)-grid are a grid-like-minor of order k +1.In [101] it is proved that every graph with treewidth Ω(k4

√log k) contains

a grid-like minor of order k. Meta-algorithmic implications of the resultsin [101], analogous to those of Theorem 1, can be found in [85].

4.2 Bidimensionality

Theorem 5 has several refinements that are important for improving theparameter dependence of the algorithm in Lemma 1. The first variant ofTheorem 5 for special graph classes appeared in [113] (proved for the twin pa-rameter of branchwidth) from which it follows that if G is a planar graph, thentw(G) ≤ 6 · gm(G). Actually, with some more careful application of the resultsof [113] it can also be proven that tw(G) ≤ 5 · gm(G), which can be improvedfurther to tw(G) ≤ 9

2 · gm(G) using the results of [62]. An analogous upperbound holds also for graphs embedded in surfaces. From the results in [27], itfollows that tw(G) ≤ 6 · (eg(G) + 1) · gm(G) where eg(G) is the Euler genus


of G. Also in [30], it was proven that if G is a K3,r-minor free graph, thentw(G) ≤ 204r · gm(G). At this point, the natural question is whether this lin-ear dependence holds for every non-trivial minor-closed graph class. This wasresolved in [29], where the following theorem has been proved.

Theorem 6. Let r be a positive integer. If G is a Kr-minor free graph, thentw(G) = Or(gm(G)).

The proof of Theorem 6 is heavily based on GMT. More specifically, it dependson the Structure Theorem of the GMT [109] which implies immense bounds forthe parameter dependence of the bound in Theorem 6. The improvement of theparameter dependance of Theorem 6 is an interesting problem and this mightbe possible without making use of the structural results of [109].

As mentioned in the previous Section, a parameter p is minor-bidimensionalif it is closed under taking of minors and for every non-negative integer k it holdsthat p(G√k) = Ω(k). A major consequence of Lemma 1, Theorem 6, and thediscussion above is the following meta-algorithmic result.

Theorem 7. Let H be an r-vertex graph and let p be a graph parameter that isminor-bidimensional and single exponentially solvable with respect to treewidth.Then k-Parameter Checking for p restricted to H-minor free graphs belongs(constructively) to 2Or(

√k) ·nO(1)-FPT, i.e., one can construct a sub-exponential

FPT-algorithm that solves it.

Notice that the above result is, in a sense, optimal, as, due to the complexitybounds in [18], a 2O(

√k) ·nO(1)-step parameterized algorithm is the best we may

expect for several bidimensional parameters, even on planar graphs. The meta-algorithmic machinery that we employed above in order to prove Theorem 7 isknown as Bidimensionality Theory and was introduced for the first time in [27],while some preliminary ideas had already appeared in [4, 52].

Theorem 7 concerns only minor-closed parameters. A typical parameter thatdoes not fit in the framework of minor-bidimensionality is the dominating setnumber, denoted by ds(G) and defined as the minimum size of a dominating setin G, i.e., a set S of vertices such that every vertex not in S has some neighborin S.

The dominating set number is not minor-closed as it may increase by removingedges. However this is not the case when we do only contractions. To developthe contraction counterpart of bidimensionality, one has to find a counterpartof Theorem 6 for contractions, i.e., to detect what types of graphs appear ascontractions in graphs with big treewidth. This line of research was developedin [26, 31] and concluded in [46]. Before we present the the results in [46], weneed first some definitions.

Let Γk (k ≥ 2) be the graph obtained from the (k × k)-grid by triangulatinginternal faces of the (k× k)-grid such that all internal vertices become of degree6, all non-corner external vertices are of degree 4, and then one corner of degreetwo is joined by edges with all vertices of the external face (the corners arethe vertices that in the underlying grid have degree two). Graph Γ6 is shown

240 D.M. Thilikos

Fig. 3. The graph Γ6

in Fig. 3. Let also Πk be the graph obtained from Γk by adding a new vertexadjacent to all vertices of Γk.

A consequence of the results in [46] is the following.

Theorem 8. There exists a function α : N → N such that every connectedgraph of treewidth at least α(k) contains some of the graphs in {Kk, Γk, Πk} asa contraction.

Theorem 8 has several refinements. One of them is the following counterpart ofTheorem 6.

Theorem 9. There exists a function β : N→ N such that every connected Kr-minor-free graph of treewidth at least β(r) · k2 contains either Γk or Πk as acontraction.

Notice that in the above theorem, the quadratic dependence (on k) is optimal.Indeed, let Zk2 be the graph obtained by adding to Gk2 a new vertex adjacentto all the k2 vertices with both coordinates in the underlying grid divisible by k.Then Zk2 excludes K6, Gk+2, and Πk+2 as contractions and is of treewidth atleast k2. This means that, in order to have a “linear counterpart” of Theorem 6,we should restrict further the graphs that we exclude. An apex graph is a graphthat can become planar by the removal of one vertex. It appears that the lineardependence in the bound of Theorem 6 is also possible for contractions whenwe consider graphs excluding some apex graph as a minor. For this, we definetgm(G) as the maximum k for which G contains Γk as a contraction.

Theorem 10. Let H be an apex graph with r vertices. If G is a connected H-minor-free graph, then tw(G) = Or(tgm(G)).

We say that a parameter p is contraction bidimensional if it is closed undertaking of contractions and if p(Γ√k) = Ω(k) for every non-negative integer


k. Using now Theorem 10 one can derive the following contraction counterpartof Theorem 7.

Theorem 11. Let H be an r-vertex apex graph and let p be graph parameterthat is contraction-bidimensional and single exponentially solvable with respectto treewidth. Then k-Parameter Checking for p restricted to H-minor freegraphs belongs (constructively) to 2Or(

√k) · nO(1)-FPT, i.e., one can construct a

sub-exponential FPT-algorithm that solves it.

The algorithmic consequence of Theorems 6 and 10 are not restricted in thedesign of sub-exponential parameterized algorithms (i.e., Theorems 7 and 11).Bidimensionality theory had meta-algorthmic applications in the automaticderivation of linear-time kernels for wide families of parameterized problems [12,51]. Apart from its applications to parameterized complexity, BidimensionalityTheory was also used for the automated design of Fast Polynomial Time Ap-proximation Schemes (FPTAS) in [28] and [48].

Proving extensions of Theorems 7 and 11 for wider families of graph classes(possibly with worse – but still moderate – time bounds) is a open challenge inparameterized algorithm design. For this, one may either need to find extensionsof Theorems 6 and 10 for graph classes that are wider than H-minor free andapex-minor free graphs respectively (see [50] for an important step in this direc-tion) or to invent alternative notions of grid-like structures whose presence in agraph is still able to certify a big enough value for the parameter p (see [85, 101]and the end of Subsection 4.1).

5 The Irrelevant Vertex Technique

One of the most powerful tools in parameterized algorithm design is the irrel-evant vertex technique, introduced in [108] in order to derive (among others)FPT-algorithms for the H-Minor Checking (Theorem 2) and the k-Disjoint

Paths Problem. The formal definition of the latter is the following.

k-Disjoint Paths

Instance: A graph G and a sequence of pairsterminals T = (s1, t1), . . . , (sk, tk) ∈ (V (G)× V (G))k.

Parameter: k.Question: Are there k pairwise vertex disjoint paths

P1, . . . , Pk in G such that for every i ∈ {1, . . . , k},Pi has endpoints si and ti?

We stress that, in [108], both H-Minor Checking and k-Disjoint Paths

where treated simultaneously and the methodology that we present below issimilar for both of them. In this section we give an outline of the Ok(n3) algo-rithm in [108] for the k-Disjoint Paths problem and we present some of themost important combinatorial results that supported the proof of its correctness.

242 D.M. Thilikos

5.1 The General Framework

Given an instance (G, T, k) of the k-Disjoint Paths problem, we say that avertex v ∈ V (G) is an irrelevant vertex of G if (G, T, k) and (G \ v, T, k) areequivalent instances of the problem.

The general scheme of the algorithm in [108] is the following:

Irrelevant Vertex for the class Gk

Input: An instance (G, T, k) of k-Disjoint Paths

Output: A (reduced) equivalent instance of k-Disjoint Paths

1. while G �∈ Gk,2. find an irrelevant vertex v in G3. set G← G \ v4. output (G, T, k)

Clearly, each variant of the above scheme depends on the parameterized classGk and creates an equivalent instance that belongs to Gk. The algorithm in [108]applies the above scheme in two phases: the first phase considers

Gk = {G | G is a Kh(k)-minor free graph}

for some recursive function h and produces equivalent instances where the inputgraph does not contain a “big clique” as a minor. The second phase assumesthat the input graph excludes such a clique and considers

Gk = {G | G is a Gg(k)-minor free graph},

for some recursive function g : N→ N. This produces an equivalent instance that,from Theorem 5, has treewidth bounded by Ok(1) and, in this case, the problemcan by solved in Ok(n) steps, using dynamic programming or, alternatively, byjust using Courcelle’s theorem.

It now remains to explain how Step 2 of the above scheme (i.e., finding anirrelevant vertex) is implemented in each of these two phases.

We omit the description of the first phase. Instead, we restrict ourselves tothe second phase, as it encompasses the most combinatorially rich part of [108].We just mention that the function h is determined from the results in [108] onthe correctness of the first phase. The function g will be defined in the course ofthe description of the second phase below.

Assume now that we have an instance (G, T, k) of k-Disjoint Paths whereG excludes a clique Kh(k) as a minor but, however, it still contains a the gridGg(k) as a minor which means that tw(G) ≥ g(k). A big part of [108] is devotedto the characterization of such graphs, i.e., of H-minor free graphs with “big”treewidth. In particular, a major achievement of [108] was to show the WeakStructure Theorem of GMT, stating that such graphs contain some portionthat is, in a sense, “almost flat”. At this point we postpone the description ofthe irrelevant vertex technique to Subsection 5.3 in order to give the precisestatement of this theorem.


5.2 The Weak Structure Graph Minors Theorem

Walls. A wall of height k, k ≥ 1, is the graph obtained from a ((k+1)×(2·k+2))-grid with vertices (x, y), x ∈ {1, . . . , 2 ·k +2}, y ∈ {1, . . . , k +1}, by the removalof the “vertical” edges {(x, y), (x, y + 1)} for odd x + y, and then the removalof all vertices of degree 1. We denote such a wall by Wk. The corners of thewall Wk are the vertices c1 = (1, 1), c2 = (2 · k + 1, 0), c3 = (2 · k + 1 + (k + 1mod 2), k+1) and c4 = (1+(k +1 mod 2), k +1). We let C = {c1, c2, c3, c4}. Asubdivided wall W of height k is a graph obtained from Wk by replacing some ofits edges by paths without common internal vertices. We call the resulting graphW a subdivision of Wk. The perimeter P of a subdivided wall is the cycle definedby its boundary. The layers of a subdivided wall W of height k are recursivelydefined as follows. The first layer of W is its perimeter. For i = 2, · · · , �k

2 �,the i-th layer of W is the (i − 1)-th layer of the subwall W ′ obtained from Wby removing from W its perimeter and all occurring vertices of degree 1 (seeFigure 4).

Fig. 4. A subdivided wall of height 5 and its two first layers. The first layer is itsboundary

Compasses and Rural Divisions. Let W be a subdivided wall in G. Let K ′ bethe connected component of G \ P that contains W \ P . The compass K of Win G is the graph G[V (K ′) ∪ V (P )]. Observe that W is a subgraph of K and Kis connected. We say that a path of K is perimetric if its endpoints lie in theperimeter P of W . Let P1 and P2 be two perimetric paths of K with endpointsa1, b1 and a2, b2 respectively. We say that P1 and P2 cross in K if (a1, a2, b1, b2)is the cyclic ordering of their endpoints in P . We say that a wall is flat in G ifK does not contain any pair of crossing and vertex-disjoint perimetric paths.

If J is a subgraph of K, we denote by ∂KJ the set of all vertices v ∈ V (J)such that either v ∈ C or v is incident with an edge of K that is not in J . Arural division D of the compass K is a collection (D1, D2, . . . , Dm) of subgraphsof K with the following properties:

1. {E(D1), E(D2), . . . , E(Dm)} is a partition of non-empty subsets of E(K),2. for i, j ∈ [m], if i �= j then ∂KDi �= ∂KDj and V (Di) ∩ V (Dj) = ∂KDi ∩

∂KDj ,

244 D.M. Thilikos

3. for each i ∈ [m] and all x, y ∈ ∂KDi there exists a (x, y)-path in Di with nointernal vertex in ∂KDi,

4. for each i ∈ [m], |∂KDi| ≤ 3, and5. the hypergraph HK = (

⋃

i∈[m]

∂KDi, {∂KDi | i ∈ [m]}) can be embedded in a

closed disk Δ such that c1, c2, c3 and c4 appear in this order on the boundaryof Δ and for each hyperedge e of HK there exist |e| mutually vertex-disjointpaths between e and C in K.

We call the elements of D flaps. A flap D ∈ D is internal if V (D) ∩ V (P ) = ∅.We can now state one of the main results in [108], known as the Weak StructureGraph Minors theorem.

Theorem 12 ([108]). There exist recursive functions g1 : N × N → N andg2 : N → N, such that for every two graphs H and G and every q ∈ N, one ofthe following holds:

1. H is a minor of G,2. tw(G) ≤ g1(q, r), where r = |V (H)|3. ∃X ⊆ V (G) with |X | ≤ g2(r) such that G \X contains as a subgraph a flat

subdivided wall W where W has height q and the compass of W has a ruraldivision D such that each internal flap of D has treewidth at most g1(r, q).

While the statement of Theorem 12 above is somehow complicated, the intuitionbehind it is simpler. It says that when a graph excludes some “small” graph H asa minor and has “big enough” treewidth, it is enough to remove a “few” verticesfrom it, i.e., the vertices in X , and take a graph G \ X where it is possibleto detect a subdivided wall W that is situated in a “flat” territory inside itsperimeter P . The part of G that is inside P is the compass K of W which canbe seen as the union of a collection of graphs (flaps) that are tree-like (havebounded treewidth) and are “planted” in that territory. Theorem 12 was usedalso in [2, 24] with the name “the Trinity Lemma”. However, a more depictivealternative nomenclature might be the “Sunny Forest Lemma”, in the sense thatthe compass K is a forest, whose trees are the flaps, and X is the sun throwingits rays at it!

In [59], an optimized version of the above result was proved where g1(r, q) =Or(q) and g2(r) is equal to the apex number of H , i.e., the minimum numberof vertices that, when removed from H , leave a planar graph. This improvedversion can easily yield both Theorems 6 and 10. In case H is an apex graph,i.e., it can become planar with the removal of a single vertex, the result in [59]implies that X = ∅ which gives an analogue of Theorem 12 for apex minor-freegraphs. As, in this case, the “sun” X does not exist, we are tempted to call this“apex”-variant of Theorem 12 the “Dark Forest Lemma”.

5.3 Irrelevant Vertices and Linkages

We now go back to the task of detecting an irrelevant vertex in a graph G thatexcludes Kh(k) and has treewidth bigger than g(k). Recall that, at this point, h


has already been determined so that the previous phase of the algorithm runscorrectly. In what follows, we set g(k) = g1(f0(k) ·f1(λ(k)), h(k)) where g1 is thefunction in Theorem 12, f0(k) = �

√2k� + 1, and f1 and λ will be determined

later.According to [108], it is possible, in Ok(n2) steps, to detect in G a set X and a

subdivided wall W of height q = f0(k) · f1(λ(k)) of G \X where |X | ≤ g2(h(k)),as indicated in Theorem 12. For simplicity, we restrict our presentation to thecase where X is an empty set, i.e., |X | = 0. Even if the ideas for the more generalcase are of the same flavor, they are quite more complicated and we prefer toomit them here.

Using a counting argument based on the definition of f0, it is easy to see thatW contains a subdivided wall W ′ of height q′ = f1(λ(k)) whose compass K ′

avoids all terminals of the pairs in T .The next step of the algorithm in [108] is based on the claim that if we take

q′ to be “big enough”, then any vertex vmid of the inner layer Lin of W ′ is anirrelevant vertex and therefore it can be safely removed from G. While such avertex is easy to detect, to proof that it is indeed irrelevant – for some suitablechoice of q′ – is not easy. We just mention that papers XXI [111] and XXII [107]of the Graph Minor series where devoted to it. Below, we present only somebasic notions and ideas used in this proof. For this, we first need the definitionof a k-linkage, introduced in [111].

A k-linkage in a graph G is a set of k pairwise disjoint paths of it. Theendpoints of a linkage L are the endpoints of the paths in L. The pattern of L isdefined as

π(L) = {{s, t} | L contains a path from s to t}

Two k-linkages are equivalent if they have the same pattern.W.l.o.g. we assume that all terminals involved in T are distinct. This implies

that every solution to the k-Disjoint Paths problem is a k-linkage, whosepattern is determined by the pairs in T . To prove the irrelevance of the vertexvmid, it is enough to show that any linkage L whose paths meet Lin can bereplaced with an equivalent one that avoids it. To obtain an idea of how pathsin L may reside inside K ′, we need to make some observations.

Let R be the linkage defined by the connected components of (⋃

L∈L L)∩K ′,i.e., the subpaths of the paths in L that are “cropped” by the compass K ′ (noticethat all paths in R are perimetric). By the flatness of W ′, it is not possible thattwo paths in R cross in K ′. Moreover, by the definition of the the rural divisionD′ of K ′, each layer of W ′, different than the inner one, is a separator of G.Therefore, if a path in R meets layers Li and Lj for i ≤ j, then it should alsomeet layer Lμ for every μ ∈ {i, . . . , j}. These observations argue that, intuitively,paths in R cross K ′ as if K ′ where a graph embedded in a disk bounded by P –see Figure 5 for a visualization of this. One may now claim that the infrastruc-ture of a “big enough” subdivided wall W ′ should provide enough space inside K ′

246 D.M. Thilikos

Fig. 5. A subdivided wall W ′ and the way a 13-linkage L is traversing its compass K′.The only vertices that are depicted are the endpoints of the paths in L (white vertices).The only edges that are depicted are those of the paths in L and the edges of W ′. Thegrey area contains the vertices and the edges of the graph G that do not belong to K′.

so that the paths of L could be rerouted to an equivalent linkage that does notenter very deeply inside K ′. To formalize this claim Robertson and Seymourdefined the notion of a vital linkage in [111].

A linkage L in a graph G is called vital if its vertices meet all the vertices ofG and if there is no other linkage in G that is equivalent to L. An example ofa vital k-linkage in a graph is depicted in Figure 6. Clearly, if a solution of thek-Disjoint Paths Problem corresponds to a vital linkage, then no irrelevantvertex can be detected. The main result of [111] asserts that this possible “lackof flexibility” of linkages vanishes when graphs have big enough treewidth.

Theorem 13. There exists a recursive function λ : N → N such that everygraph with a vital k-linkage has treewidth at most λ(k).

Actually, it was also proved in [111] that treewidth can be replaced by pathwidthin Theorem 13. As the proof of 13 uses the Structure Theorem of the GMT [109],the upper bound for λ that follows from [111] is immense. However it was provedin [3] that in the case of planar graphs it holds that λ(k) = 2O(k). Moreover, thisbound is, in a sense, tight: as argued in [3], for each k it is possible to constructa planar graph that contains a vital k-linkage and has treewidth 2Ω(k) (the 5-linkage in the graph of Figure 6 already gives the flavor of such a construction).

Let now G′ be the subgraph of G defined by the union of the paths in L, andthe compass K ′ of W ′. At this point, a naive idea might be to directly applyTheorem 13 and set q′ = λ(k) so that the linkage L of G′, corresponding to asolution of the k-Disjoint Paths problem, cannot be vital. However, from thisalone, we cannot expect nothing better than avoiding some vertices that will not


Fig. 6. A graph of treewidth 17 and a vital 5-linkage in it

necessarily be the vertices in Lin. Therefore, a non-vital linkage alone does notprovide the flexibility we need in order to reroute in G′ the paths of L in a waythat Lin is avoided.

Curiously, it appears that the importance of non-vital linkages is rather qual-itative than quantitative. Based on their “elementary” flexibility, it is possibleto prove that none of the paths in R “bounces” much. In particular, L can bechosen in a way that if a path in R meets some layer Li in two different verticesx and y, then its subpath between x and y will not meet any layer Lj where|i − j| ≥ f1(λ(k)), for some recursive function f1. This directly implies thatpaths in R do not go deeper than layer Lf1(λ(k)) and thus they avoid Lin whenq′ = f1(λ(k)). That way, it is possible to prove what we need: if the height ofW ′ is f1(λ(k)), then another linkage, equivalent to L exists in G′ (and thereforein G as well) that avoids Lin.

We should stress that even if the above sketch might be “convincing” for agood-tempered reader, it is far from being a formal proof. In the more realisticcase where X is non-empty, a more complicated criterion for the choice of thesubdivided wall W ′ should be devised and a bigger lower bound for the heightof W ′ is necessary so that it contains an irrelevant vertex. In fact, this requiresbigger lower bounds for both f0 and f1. The whole proof is quite technical andhas been the main purpose of [107].

According to the above discussion, the second phase of the algorithm runsin Ok(n3) steps and outputs a graph of treewidth at most g(k). As provedin [116], the k-Disjoint Paths problem can be solved by a f2(k)·n step dynamicprogramming algorithm where f2(k) = 2O(k log k) (see [1, 90] for results relatedto this problem). As the parameter dependence of the running time of this laststep is dominant in the running time of the algorithm, we conclude that theoverall parameter dependence is:

O(f2(g1(f0(k) · f1(λ(k)), h(k)))).

Clearly, an improvement on the existing bounds for any of the functionsg1, h, f0, f1, f2, and λ would be an important step towards reducing the

248 D.M. Thilikos

parameter dependence of the algorithm for the k-Disjoint Paths problem.In fact, the only function that is “really immense” is λ, because the proof of itsexistence was based on the Structure Theorem of the GMT [109]. In this direc-tion an alternative, relatively simpler, proof was given in [80] that avoids thecore results of [109]. Using a rough estimation, the proof in [80] should give that

that λ(k) = 222O(k)

which changes the status of the parameter dependence inRoberson and Seymour’s algorithm from “immense” to “huge”. Clearly, any fur-ther improvements, even for special cases or variants of the problem, are highlywelcome (see [3]).

5.4 Applications

The above description already outlines a powerful algorithmic framework thatcould not be of use for just one problem. Below, we mention a series of results inparameterized algorithms where the irrelevant vertex technique (or extensionsof it) has been applied. We sort them in chronological order of their appearance.

[24] A proof of the following meta-agorithmic result: Let C be a class of graphsexcluding and h-vertex graph H as a minor. Then any first-order defin-able decision problem can be solved in time Oh+|φ|(nO(1)), where f is acomputable function and φ is the sentence defining the decision problem.

[77] A 2O(g) ·n step algorithm that, given a graph G and a non-negative integerg either outputs an embedding of G in a surface of genus g or a minor ofG that belongs to obs≤m(Gg) where Gg contains all graphs embeddible ina surface of Euler genus g. A previous result of this type, but not withsingle-exponential parameter dependence appeared previously in [94].

[2] A proof that it is possible to construct an algorithm that, given the ob-struction sets of two minor-closed graph classes G1 and G2, outputs theobstruction set of the class G = G1 ∪ G2. Also, in the same paper, it wasproved that it is possible to construct an algorithm that given the ob-struction set of a minor-closed graph class G and a non-negative integerk, outputs the obstruction set of the class k-almost(G) = {G | ∃S ⊆V (G) : |S| ≤ k and G \ S ∈ G} (see also [19, 71, 93] for related results).

[83] An Ok(nO(1)) time algorithm for solving the Induced Cycle Through

Terminals problem: Given a graph G, embedded in some surface, and aset S ⊆ V (G) of terminals, does G contain an induced cycle that meetsall vertices in S?

[60] An 2O(k3/2) ·nO(1) time algorithm for the Odd Induced Cycle Packing

on planar graphs: Given a graph G and an integer k, does G contains kinduced odd cycles?

[79] An Ok(nO(1)) time algorithm for the Odd Cycle Packing problem:Given a graph G and an integer k, does G contains k vertex-disjoint oddcycles?


[64] An Ok(n) time algorithm for the Bipartite Contraction problem:Given a graph G and an integer k, can we obtain a bipartite graph fromG by a sequence of at most k edge contractions in G?

[49] Subexponential 2O(√

k)·n time algorithms for the PartialVertex Cover

and PartialDominatingSet problems for apex minor-free graphs:Givena graph G and integers k, t, can we cover (resp. dominate) at least t edges(resp. vertices) with at most k vertices?

[61] An Ok(n3) algorithm for the Topological Minor Containment andthe Immersion Containment problems: Given two graphs G and H ,where n(H) = k, does G contain H as a topological minor (resp. immer-sion). The results in [61] can be seen as a major extension of the algorithmin [108].

[45] A proof of the following result on kernelization: Let Gr be the class ofall Kr-minor free graphs. Then the Dominating Set Problem and theConnected Dominating Set problem, asking whether a graph G has a(connected) dominating set of size k, has a linear Or(k)-size kernel whenrestricted in graphs in Gr.

[69] An Ok+g(n3) algorithm for the Contraction Containment problemrestricted to graphs of Euler genus g. The Contraction Containment

problem asks, with input two graphs G and H , where n(H) = k, whetherH is a contraction of G.

Clearly, the above list is just indicative and is expected to grow more. Furtheralgorithmic applications of the weak structure theorem and/or the irrelevantvertex technique can be found in [66, 68, 70, 72, 73, 75, 76, 78]. Also resultswhere the irrelevant vertex idea is applied an a more general sense, withoutusing directly results of the GMT, can be found in [25, 47, 63, 92].

6 Conclusions

Covering the whole range of the contributions of the GMT to the design ofparameterized algorithms is a task that cannot fit in the space of this shortpresentation. The progress over the last years towards building an AlgorithmicGraph Minors Theory has been noticeable and we believe that there is much more“algorithmic material” to be extracted from this deep and fascinating theory.As we expect more results to emerge from GMT, not only in parameterizedalgorithms by also in other fields of algorithm design, we hope that this smallportion of the material covered will be of use as in invitation to this direction.

Acknowledgements. We wish to thank Isolde Adler, Marcin Kaminski andStavros G. Kolliopoulos for their detailed comments, remarks, and suggestionson this text. We also wish to personally thank Michael R. Fellows for his supportand encouragement while investigating this “minor” corner of parameterizedalgorithms.

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